would the derivative of |_x_| be equal to 1? Or is it different because of the brackets?

Is the function even continuous? If you look at the function $\displaystyle fn,n+1)\mapsto\mathbb{R},\text{ }n\in\mathbb{N}$ with $\displaystyle f(x)=\left \lfloor x\right\rfloor$ then in fact $\displaystyle f(x)=1$. And so $\displaystyle f'(x)=0$.